Section 11.2:

1. Section 11.2: Systems of Linear Equations in Three Variables

2. 11.2 Lecture Guide: Systems of Linear Equations in Three Variables Objective: Solve a system of three linear equations in three variables.

3. 1. Determine whether is a solution of

4. Types of Solution Sets for Linear Systems with Three Equations can have The linear system An Infinite Number of Solutions One Solution No Solution The planes have no point in common; the system is inconsistent. The planes intersect along a line and thus have an infinite number of common points; the system is consistent and the equations are dependent. The planes intersect in a single point P; the system is consistent and the equations are independent.

5. Strategy for Solving a System of Linear Equations* Step 1. Write each equation in the general form Ax +By + Cz =D. Step 2. Select one pair of equations and use the substitution or the addition method to eliminate one of the variables. Step 3. Repeat Step 2 with another pair of equations. Be sure to eliminate the same variable as in Step 2. Step 4. Eliminate another variable from the pair of equations produced in Steps 2 and 3 and solve this system of equations. Step 5. Back-substitute the values from Step 4 into one of the original equations to solve for the third variable. Step 6. Does this solution check in all three of the original equations?

6. *If a contradiction is obtained in any of these steps, the system is inconsistent and has no solution. If an identity is obtained in any step, the system is either dependent with infinitely many solutions or inconsistent with no solution.

7. 2. Solve the System of Linear Equations: (a) Produce a system of equations.

8. 2. Solve the System of Linear Equations: (b) Produce an equation with only one variable.

9. 2. Solve the System of Linear Equations: (c) Back-substitute to determine the values of the remaining variables.

10. 2. Solve the System of Linear Equations: (d) Write the answer as an ordered triple.

11. Properties of the Reduced Echelon Form of a Matrix 1. The first nonzero entry in a row is a 1. All other entries in the column containing the leading 1 are zeros. 2. All nonzero rows are above any rows containing only zeros. 3. The first nonzero entry in a row is to the left of the first nonzero entry in the following row.

12. Transforming an Augmented Matrix into Reduced Echelon Form Step 1. Transform the first column into this form by using the elementary row operations to (a) produce a 1 in the top position and (b) use the 1 in row 1 to produce zeros in the other positions of column 1.

13. Transforming an Augmented Matrix into Reduced Echelon Form Transform the next column, if possible, into this form by using the elementary operations to Step 2. • (a) produce a 1 in the next row and (b) use the 1 in this row to produce zeros in the other positions of this column. If it is not possible to produce a 1 in the next row, proceed to next column.

14. Transforming an Augmented Matrix into Reduced Echelon Form Step 3. Repeat Step 2 column by column, always producing the 1 in the next row, until you arrive at the reduced form.

15. 3. Write an augmented matrix for the system of linear equations.

16. 4. Write a system of linear equations in x, y, and z that is represented by the augmented matrix.

17. 5. Complete the steps to solve the following system of linear equations.

18. 5. Complete the steps to solve the following system of linear equations. Solution: ____________

19. 6. Label the row operations used to solve the following system of linear equations. Then write the solution for this system.

20. 6. Label the row operations used to solve the following system of linear equations. Then write the solution for this system. Solution: ____________

21. Enter the matrix associated with each system into your calculator and use the rref feature to solve the system. Give the reduced form of each matrix and the solution of each system. See Calculator Perspective 11.2.1. 7. Reduced form: Solution: ____________ Does this solution check in all three equations?

22. Enter the matrix associated with each system into your calculator and use the rref feature to solve the system. Give the reduced form of each matrix and the solution of each system. See Calculator Perspective 11.2.1. 8. Reduced form: Solution: ____________ Does this solution check in all three equations?

23. Forms That Indicate a Dependent or Inconsistent n x n System of Equations Let A be the augmented matrix of an n x n system of equations. 1. If the reduced form of A has a row of the form where then the system is inconsistent and has no solution. 2. If the system is consistent and the reduced form of A has a row of the form (all zeros), then the system is dependent and has infinitely many solutions.

24. 9. The following system is an inconsistent system. Note the contradiction that occurs in the reduced form of the matrix. Enter the matrix associated with the system into your calculator and use the rref feature to solve the system. Give the reduced form of the matrix and the solution of the system. Reduced form: Solution:

25. 10. The following system is a consistent system of dependent equations. Note the identity that occurs in the reduced form of the matrix. Enter the matrix associated with the system into your calculator and use the rref feature to solve the system. Give the reduced form of the matrix and the solution of the system. Give both the general solution and two particular solutions. Reduced form: General Solution: _______________________ Particular Solution #1: ____________ Particular Solution #2: ____________

26. 11. The sum of three numbers is 105. The third number is eleven less than ten times the second number. Two times the first number is 7 more than three times the second number. What are the numbers?